If you're looking to find differences between two folders look at WinMerge. This is a diff tool primarily intended for dealing with source code. It doesn't really treat one folder as old and one folder as new, it just looks for any differences between files in the folders (and missing/new files).

If you're looking for a backup tool try Syncback. They have a freeware version and also a paid version. This can keep directories in sync or perform a backup of one directory to another. It isn't really intended to show you the actual differences, although if you run a simulated backup you can get a list of the files that would be affected by the backup (ie: new or changed files).